MHB 231.13.3.75 Find the center of R

[karush]

$\tiny{231.13.3.75}$
$\textsf{Imagine $3$ unit spheres (radius equal to 1) }$
$\textsf{with centers at, $o(0,0,0)$, $p(\sqrt{3},-1,0)$ and $q(\sqrt{3},1,0)$.} \\$
$\textsf{Now place another unit sphere symmetrically on top of these spheres with its center at R.} \\$
$\textsf{a Find the center of R.} \\$
$\textsf{b. Let $r_\eta$ be the vector from the center
of the sphere $i$ to the center of sphere $j$} \\$

$\textsf{Find $\displaystyle r_{op} , r_{oq}, r_{pq}, r_{or} , $ and $r_{pr}$} $

$\textit{ok, first, I don't think I understand what this looks like ??}$

 

[Prove It]

$\tiny{231.13.3.75}$
$\textsf{Imagine $3$ unit spheres (radius equal to 1) }$
$\textsf{with centers at, $o(0,0,0)$, $p(\sqrt{3},-1,0)$ and $q(\sqrt{3},1,0)$.} \\$
$\textsf{Now place another unit sphere symmetrically on top of these spheres with its center at R.} \\$
$\textsf{a Find the center of R.} \\$
$\textsf{b. Let $r_\eta$ be the vector from the center
of the sphere $i$ to the center of sphere $j$} \\$

$\textsf{Find $\displaystyle r_{op} , r_{oq}, r_{pq}, r_{or} , $ and $r_{pr}$} $

$\textit{ok, first, I don't think I understand what this looks like ??}$

It is three spheres arranged into an equilateral triangle, with a fourth sphere placed on top to form a tetrahedron...

 

[karush]

so then we can assume the distance between all the centers is =. (or magnitude)

$R\left(\frac{\sqrt{3}}{2},0,\frac{\sqrt{3}}{2}\right)$